Bimodues associated to twisted modules of vertex operator algebras and fusion rules

TL;DR

This paper constructs bimodules associated with twisted modules of vertex operator algebras and investigates their role in understanding fusion rules, providing new algebraic tools for analyzing module interactions.

Contribution

It introduces a new bimodule construction linked to twisted modules and explores its applications to fusion rule analysis in vertex operator algebras.

Findings

Construction of an $A_{g_1g_2}(V)$-$A_{g_2}(V)$-bimodule

Connection between bimodules and fusion rules

Enhanced understanding of module interactions in twisted settings

Abstract

Let $V$ be a vertex operator algebra, $T\in \mathbb{N}$ and $(M^k, Y_{M^k})$ for $k=1, 2, 3$ be a $g_k$-twisted module, where $g_k$ are commuting automorphisms of $V$ such that $g_k^T=1$ for $k=1, 2, 3$ and $g_3=g_1g_2$. Suppose $I(\cdot, z)$ is an intertwining operator of type $({array}{c} M^{3} M^{1} M^{2} {array}) $. We construct an $A_{g_1g_2}(V)$-$A_{g_2}(V)$-bimodule $A_{g_1g_2, g_2}(M^1)$ which determines the action of $M^1$ from the bottom level of $M^2$ to the bottom level of $M^3$ and explored its connections with fusion rules.